Prove that if a divides bn and a,b are relatively prime then a divides n

For a natural number , the Fundamental Theorem of Arithmetic claims that where all are primes (some primes may occur several times). In such case, let denote the multiset. The Fundamental Theorem also says that for each , is uniquely defined. E.g., .

It is clear that . (Indeed, the right-hand side is one factorization of ; therefore, it is the only one.) Also, is a divisor of if and only if . The right-to-left direction is obvious, and if for some , then , so .

Finally, it is easy to show that if and only if .

All right, assume that divides and . Then and . Therefore, , i.e., divides .

Maybe this is not the "classical" and the simplest proof, but it should work.